Lagrangian Inference for Ranking Problems

We propose a novel combinatorial inference framework to conduct general uncertainty quantification in ranking problems. We consider the widely adopted Bradley-Terry-Luce (BTL) model, where each item is assigned a positive preference score that determines the Bernoulli distributions of pairwise comparisons' outcomes. Our proposedmethod aims to infer general ranking properties of the BTLmodel. The general ranking properties include the local properties such as if an item is preferred over another and the global properties such as if an item is among the top K-ranked items. We further generalize our inferential framework to multiple testing problems where we control the false discovery rate (FDR) and apply the method to infer the top-K ranked items. We also derive the information-theoretic lower bound to justify the minimax optimality of the proposed method. We conduct extensive numerical studies using both synthetic and real data sets to back up our theory.

Automated summary

We propose a novel combinatorial inference framework for uncertainty quantification in ranking problems. Our method considers the widely adopted Bradley-Terry-Luce (BTL) model and aims to infer general ranking properties, including local and global properties. We also extend the framework to multiple testing problems and derive an information-theoretic lower bound. Extensive numerical studies using synthetic and real datasets support our theory.